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When did Neil Armstrong set foot on the Moon?
Short answer
Neil Armstrong set foot on the Moon at 02:56:15 UT on July 21, 1969, already early morning in Europe, with the Eagle lunar module having landed at 20:17:40 UT the day before. The moonwalk ended at 5:11:13 UT, when he and Aldrin returned to the module and closed the hatch.

The hour and day almost everyone misremembers
The exact figure is 02:56:15 UT on July 21, 1969. Half the planet remembering July 20 has an explanation: that was the day of the lunar landing, at 20:17:40 UT, and also the local date in the Americas when Armstrong climbed down the ladder. The Eagle had been sitting in the Sea of Tranquility for several hours before the hatch opened, because the surface timeline included checks, suit preparation and cabin depressurization before going outside.
What happened in those minutes
Armstrong climbed down the ladder, stepped onto the regolith and spoke the line that the annotated transcript records as "That's one small step for (a) man; one giant leap for mankind." He then described the terrain with the precision of a report: fine and powdery, able to be scuffed up with the toe of his boot, clinging in thin layers to the sole and sides of his footwear. Buzz Aldrin came down afterward, and the two worked together during a single excursion that ended at 5:11:13 UT.
What came next
The Eagle lifted off from the Moon at 17:54:01 UT on July 21, after 21 hours and 36 minutes on the surface, to rejoin the command module Columbia, where Michael Collins had kept orbiting the whole time. The three splashed down in the Pacific on July 24, 1969, at 16:50:35 UT with 21.55 kg of lunar rock and soil. Armstrong, the first to step onto the surface, was also the first to climb back in—just a few hours on the ground that forever divided before and after.
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